Interpolation theorem between \(B_0^p\) and BMO (Q2747296)

Let \(1<p<\infty\), and let NEWLINE\[NEWLINE B^p=\left\{ f\in L^p_{\text{loc}}({\mathbb{R}}^n): \|f\|_{B^p}=\sup_{R\geq 1} \left ({1\over |B(0,R)|}\int_{B(0,R)}|f(x)|^p dx\right)^{1/p}<\infty\right\} NEWLINE\]NEWLINE where \(B(0,R)\) is the open ball in \({\mathbb{R}}^n\) centered at \(0\) with radius \(R>0\), and NEWLINE\[NEWLINE B^p=\left\{ f\in B^p({\mathbb{R}}^n): \lim_{R\rightarrow \infty} \int_{B(0,R)}|f(x)|^p dx=0\right\}. NEWLINE\]NEWLINE Let NEWLINE\[NEWLINE A^p=\left\{ f: \|f\|_{A^p}=\inf_{\omega\in \Omega}\left( \int_{{\mathbb{R}}^n}|f(x)|^p \omega(x)^{1-p} dx\right)^{1/p}<\infty\right\}, NEWLINE\]NEWLINE where \(\Omega\) is the class of functions \(\omega\) on \({\mathbb{R}}^n\) such that \(\omega\)'s are positive, radial, nonincreasing with respect to \(|x|\), and NEWLINE\[NEWLINE \omega(0)+\int_{{\mathbb{R}}^n}\omega(x) dx=1. NEWLINE\]NEWLINE For \(1<p<\infty\), we define the Hardy space \(HA^p\) associated to \(A^p\) with a norm \(\|\cdot\|_{HA^p}\) as follows NEWLINE\[NEWLINE HA^p=\{ f\in A^p: f\text{ real },\;f^*\in A^p\} NEWLINE\]NEWLINE where \(f^*\) is the non-tangential maximal function of the Poisson integral of \(f\). In this paper, the author proves that the dual of \(HA^p\) is the space \(\text{CMO}^{p^\prime}\), where \(1<p<\infty\) and \({1\over p}+{1\over p^\prime}=1\). Here \(\text{CMO}^p\) is the space of functions of central mean oscillation of order \(p\), i.e., NEWLINE\[NEWLINE \|f\|_{\text{CMO}^p}=\sup_{R\geq 1}\left ( {1\over |B(0,R)|}\int_{B(0,R)}|f(x)-m_R(f)|^p dx\right)^{1/p}<\infty, NEWLINE\]NEWLINE where NEWLINE\[NEWLINE m_R(f)={1\over |B(0,R)|}\int_{B(0,R)}f(x) dx. NEWLINE\]NEWLINE The authors also proves that the Hardy-Littlewood maximal operator is bounded on the space \(B^p\). More precisely, there exists a constant \(C_p\) depending on \(n\) and \(p\) only such that NEWLINE\[NEWLINE \|M(f)\|_{B^p}\leq C_p\|f\|_{B^p}.NEWLINE\]